E=mc² - A Note On Units

In my last post, I talked about some of the problems A Level Physics students encounter in mass-energy (E=mc²) calculations. In passing, I mentioned use of unfamiliar units during the calculations. This is a guide to some of the most common units used in the calculations.

Mass-energy equivalence calculations use the iconic equation E=mc². The equation E=mc² is often written as

ΔE = c²Δm

Where:

ΔE = Energy released in Joules (J)

Δm = Mass defect (Decrease in TOTAL particle ) in kilograms (kg)

c² = a constant equal to the speed of light squared (≈ 3 x 10¹⁶)

Joules and kilograms are the SI units for energy and mass, respectively. However, A Level Physics students familiar with mass-defect questions will have noticed that these questions rarely use joules or kilograms as units. That’s because the energy released and the mass defect in a single nuclear event (decay, fission, fusion) are very, very small quantities which involves tiny numbers if calculated in joules or kilograms.

More often, A Level Physics questions on mass-defect and binding energy use Atomic Mass Units (a.m.u. or just u for short) for mass and MeV (mega-electronvolts) for energy. Atomic mass units are, as the name suggests, the units used for the atomic mass of atomic nuclei, where protons and neutrons each have a mass of approximately 1u. For mass-energy calculations, very precise values for mass are used: often 5 or 6 significant figures (s.f.), precise enough to record different values for protons and neutrons. The “conversion factor” between a.m.u. and MeV is not c² but 931.5. That is to say, 1u is roughly equivalent to 931.5MeV.

The other set of units student may encounter are MeV for energy and MeV/c² for mass. This is simply a set of units which incorporates the constant c² into the unit for mass itself so that the numbers for equivalent mass and energy are the same. E.g. an electron which has a rest energy of 0.511 MeV will have a rest mass of 0.511 MeV/c² . Think of filling a car with petrol: we almost never pump a particular volume of petrol measured in litres or gallons. Instead, you pump £20 worth of petrol or £30 or £50. The volume of petrol which we have pumped has been measured in terms of it’s equivalent cost in pounds. So if we ask “what is the cost of £50 worth of petrol?” the answer is obvious: £50 because we pumped that particular volume in order that it cost exactly £50. Similarly, if the rest mass of an electron is 0.511 MeV/ c² its rest energy will be 0.511 MeV since the mass has been calculated here according to the equivalent rest energy.

In my last post, I talked about some of the problems A Level Physics students encounter in mass-energy (E=mc²) calculations. In passing, I mentioned use of unfamiliar units during the calculations. This is a guide to some of the most common units used in the calculations.

Mass-energy equivalence calculations use the iconic equation E=mc². The equation E=mc² is often written as

ΔE = c²Δm

Where:

ΔE = Energy released in Joules (J)

Δm = Mass defect (Decrease in TOTAL particle ) in kilograms (kg)

c² = a constant equal to the speed of light squared (≈ 3 x 10¹⁶)

Joules and kilograms are the SI units for energy and mass, respectively. However, A Level Physics students familiar with mass-defect questions will have noticed that these questions rarely use joules or kilograms as units. That’s because the energy released and the mass defect in a single nuclear event (decay, fission, fusion) are very, very small quantities which involves tiny numbers if calculated in joules or kilograms.

More often, A Level Physics questions on mass-defect and binding energy use Atomic Mass Units (a.m.u. or just u for short) for mass and MeV (mega-electronvolts) for energy. Atomic mass units are, as the name suggests, the units used for the atomic mass of atomic nuclei, where protons and neutrons each have a mass of approximately 1u. For mass-energy calculations, very precise values for mass are used: often 5 or 6 significant figures (s.f.), precise enough to record different values for protons and neutrons. The “conversion factor” between a.m.u. and MeV is not c² but 931.5. That is to say, 1u is roughly equivalent to 931.5MeV.

The other set of units student may encounter are MeV for energy and MeV/c² for mass. This is simply a set of units which incorporates the constant c² into the unit for mass itself so that the numbers for equivalent mass and energy are the same. E.g. an electron which has a rest energy of 0.511 MeV will have a rest mass of 0.511 MeV/c² . Think of filling a car with petrol: we almost never pump a particular volume of petrol measured in litres or gallons. Instead, you pump £20 worth of petrol or £30 or £50. The volume of petrol which we have pumped has been measured in terms of it’s equivalent cost in pounds. So if we ask “what is the cost of £50 worth of petrol?” the answer is obvious: £50 because we pumped that particular volume in order that it cost exactly £50. Similarly, if the rest mass of an electron is 0.511 MeV/ c² its rest energy will be 0.511 MeV since the mass has been calculated here according to the equivalent rest energy.